Question

A block of mass 0.5 kg is pushed against a horizontal spring until the spring is compressed a distance x = 0.4 m. The spring constant is 450 N/m. When it is released the block travels along a horizontal frictionless surface then reaches a vertical circular track of radius R = 2.9 m and continues to move up the track. The block experiences friction while sliding up the track. The block reaches the top of the track with a speed of 4.1 m/s. Find the change in mechanical energy caused by friction (in J).

Answer 1

Answer:3J

Explanation: friction makes changes in velocity,

So, ∆M.E = ∆K.E

∆K.E = K.E when no friction - K.E when there is friction .

To find velocity at top (no friction),

At top always , centripetal force = weight

i.e, mv^2/r = mg

We get v= √29

Velocity when there's friction = 4.1(given)

Now substitute everything.

∆K.E= 1/2m{(√29)^2-(4.1)^2}= 12/4=3J

Answer 2

Concept:- Here we have to use law of conservation of energy,

Solution:-When the spring is compressed,upto 0.4m,the work that is done in compressing it is converted into potential energy of spring...

When the spring expands then it transfer it's total energy ,to block attached to it

in that case

let the velocity gained by block be v

now,from above concept we can say

$\frac{1}{2}k {x}^{2} = \frac{1}{2}m {u}^{2} \\ 450 \times .16 = 0.5 \times {u}^{2} \\ u = 12$

total energy at the bottom=total energy at the top

$\frac{1}{2} \times m {u}^{2} + 0 = \frac{1}{2}m {v}^{2} + mg \times 2r$

here,u=velocity at the bottom and v=velocity at the top

$\frac{1}{2}m ({u}^{2} - {v}^{2}) = mg \times 2 \times 3$

$144 - {v}^{2} = 120 \\ v = 2 \sqrt{6} = 4.8 = 5(approx)$

but it is given that, velocity at the top is 4.1m/s or 4(approx)

Mechanical energy at the top that should be without friction=

$\frac{1}{2} \times 0.5 \times 25 + .5 \times 10 \times 6 \\ = 6.25 + 30 = > 36.25$

Real mechanical energy=

$\frac{1}{2} \times 0.5 \times 16 + 0.5 \times 10 \times 2r \\ 4 + 30 \\ = 34$

so,change in mechanical energy due to friction=

$36 - 34 = 2</strong><strong>j</strong><strong>$

{hope it helps you}